Control of Redundant Laser Processing Machines

ABSTRACT

A method controls an operation of a laser processing machine with redundant actuators including a first actuator and a second actuator. The method determines a feasible region for states of the first actuator and states of a reference trajectory of the first actuator defined by constraints of the laser processing machine, constraints on the reference trajectory and constraints on a range of motion of the second actuator. The method selects a subset of the feasible region, such that for any state of the first actuator and any state of the reference trajectory within the subset, there is an admissible control maintaining the state of the first actuator within the subset of the feasible region for admissible future states of the reference trajectory, and selects an admissible control action for controlling the operation such that the state of the first actuator remains in the subset of the feasible region.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the priority under 35 U.S.C. §119(e) from U.S.provisional application Ser. No. 61/912,599 filed on Dec. 6, 2013, thedisclosure of which is being incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to controlling laser processingmachines, and more particularly to controlling laser processing machineswith redundant actuators.

BACKGROUND OF THE INVENTION

A laser processing machine with redundant actuators includes multipleactuators for moving a position of a laser beam along one direction.Thus, the laser beam is over-actuated, and degrees of freedom areavailable to optimize the actuation of the laser beam along a desiredprocessing pattern. For example, the laser beam can be positioned byindependent operations of the redundant actuators, and thus the task ofpositioning the laser beam along the processing pattern can be separatedbetween redundant actuators. A reference trajectory for each redundantactuator should be generated such that the combined motion of theactuators results in the laser beam tracking the processing pattern.

Some conventional methods, see, e.g., U.S. Pat. No. 5,452,275, U.S. Pat.No. 5,798,927, U.S. Pat. No. 5,751,585, U.S. Pat. No. 6,706,999, usefrequency separation techniques to assign the task of positioning thelaser beam to two actuators. For example, the processing pattern isfiltered by a low pass filter. The filtered signal becomes a referencetrajectory for one actuator, while a difference between the processingpattern and the filtered signal becomes a reference trajectory foranother actuator. However, the filtering does consider variousconstraints of the actuators, such as constraints on the accelerationsor velocities. Furthermore, there is no guarantee that the separation infrequencies provides the optimal reference trajectories.

One method described in U.S. Publication 2013/0190898 generates thereference trajectories that account for the constraints based on ModelPredictive Control (MPC). MPC is based on an iterative, finite horizonoptimization of a model of a machine and has the ability to anticipatefuture events to take appropriate control actions. This is achieved byoptimizing the operation of the machine over a future finitetime-horizon subject to constraints, and only implementing the controlover the current timeslot. For example, the constraints can representphysical limitation of the machine, legitimate and safety limitations onthe operation of the machine, and performance limitations on atrajectory. A control strategy for the machine is admissible when themotion generated by the machine for such a control strategy satisfiesall the constraints.

For example, at a current time t, a state of the machine is sampled andan admissible cost minimizing control strategy is determined for arelatively short time horizon in the future. Specifically, an online oron-the-fly calculation determines a cost-minimizing control strategyuntil a future time t+T. Only the first step of the control strategy isimplemented, then the state is sampled again and the calculations arerepeated starting from the new current state, yielding a new control andnew predicted state path. The prediction horizon is continuously shiftedforward. For this reason, MPC is also called receding horizon control.

However, due to the tracking nature of the MPC in this problem, such areceding horizon control approach has no guarantee, in general, offinding a solution to the optimization problem. Due to the recedinghorizon nature of the finite horizon optimal control problem, theexistence of the solution for a certain window of data (horizon) doesnot by itself guarantees that when the data window are shifted, asolution still exists.

Accordingly, there is a need for a method for controlling an operationof a laser processing machine with redundant actuators that guarantees apriori satisfaction of constraints of the operation for variety ofprocessing patterns.

SUMMARY OF THE INVENTION

It is an objective of some embodiments of the invention to provide asystem and a method for controlling a constrained operation of a laserprocessing machine according to a processing pattern. It is anotherobjective to provide such a system and a method that guarantees that atany moment of the controlling there is a reference trajectory resultingin a feasible state of the machine satisfying the constraints of theoperation for all possible future variations of the referencetrajectory.

Various embodiments of the invention transform the tracking problem intoan optimization problem subject to constraints. For example, someembodiments of the invention are based on a realization that theprocessing pattern can be modeled by a dynamical system driven by abounded unpredictable input. Such realization allows defining theproblem as the control of a dynamical system to track a referencetrajectory generated by another uncertain bounded dynamical system. Theconstraints on the input of the reference trajectory can reduce the needfor determining the entire reference trajectory in advance, which can beadvantageous for some controller with limited computation capabilities.

Some embodiments of the invention are based on another realization thata rate of change in time of the reference trajectory can be consideredas a design parameter. Given space-based processing pattern, atime-based a reference trajectory can be determined to satisfy the rateof change of the reference trajectory.

Some embodiments of the invention are based on another realization thatin order to correctly track the processing trajectory be the laserprocessing machine having redundant actuators, a reference trajectory ofa first actuator has to track the processing pattern with a trackingerror bounded by the operating range of the second actuator, althoughthe tracking error does not need to be zero. In those applications, thetracking error on the reference trajectory can also be represented as aconstraint to be satisfied.

The constraints of the machine, constraints on the transient of thereference trajectory and constraints on bounds of a tracking error forma feasible region of a state of the laser processing machine and a stateof the reference trajectory. Accordingly, it is possible to optimize thecontrol maintaining the state of the machine and the state of thereference trajectory within that feasible region and, thus, to convertthe tracking problem into purely optimization problem.

Due to the nature of optimization-based receding horizon control, theexistence of a solution for a certain horizon does not by itselfguarantees the existence of the solution for a subsequent horizon.However, some embodiments of the invention are based on yet anotherrealization that it is possible to select a subset of the feasibleregion, such that from any state of the laser processing machine and ofthe reference trajectory within that subset, there is a controlmaintaining the state of the machine within the subset. Accordingly, ifa cost function representing the operation of the machine is optimizedsubject to constraints defined by that special subset of the feasibleregion, as contrasted with the optimization within the feasible regionitself, there is a guarantee that the resulting optimal trajectorytracks the reference trajectory with the bounded error, and is alwaysfeasible.

Accordingly, one embodiment discloses a method for controlling anoperation of a laser processing machine with redundant actuatorsincluding a first actuator and a second actuator. The method includesdetermining a feasible region for states of the first actuator andstates of a reference trajectory of the first actuator defined byconstraints of the laser processing machine, constraints on thereference trajectory and constraints on a range of motion of the secondactuator; selecting a subset of the feasible region, such that for anystate of the first actuator and any state of the reference trajectorywithin the subset, there is an admissible control maintaining the stateof the first actuator within the subset of the feasible region foradmissible future states of the reference trajectory determined byaccording to a model of the reference trajectory and the constraints ofthe reference trajectory; and selecting an admissible control action forcontrolling the operation such that the state of the first actuatorremains in the subset of the feasible region. The steps of the methodare performed by a processor.

Another embodiment discloses a controller for controlling an operationof a first actuator of a laser processing machine according to areference trajectory by selecting a control action for controlling theoperation such that a state of the first actuator for admissible futurestates of the reference trajectory remains in a subset of the feasibleregion of states of the first actuator and states of the referencetrajectory, wherein the feasible region is defined by constraints of thefirst actuator, constraints on the reference trajectory and constraintson a range of motion of a second actuator of the laser processingmachine, and wherein the subset of the feasible region is a controlinvariant, such that for any state of the first actuator within thesubset, there is a control maintaining the state of the first actuatorwithin the subset for all admissible future states of the referencetrajectory.

Yet another embodiment discloses a method for controlling an operationof a laser processing machine according to a processing pattern, whereinthe laser processing machine includes a first actuator and a secondactuator. The method includes determining a time-based referencetrajectory of the first actuator tracking the processor pattern with anerror bounded by a range of motion of the second actuator of the laserprocessing machine, such that a control action that changes a state ofthe first actuator according to the reference trajectory maintains thestate of the first actuator within a subset of the feasible region ofstates of the first actuator and states of the reference trajectory,wherein the feasible region is defined by constraints of the firstactuator, constraints on the reference trajectory and constraints on therange of motion of the second actuator of the laser processing machine,and wherein the subset of the feasible region is a control invariant,such that for any state of the first actuator within the subset, thereis a control maintaining the state of the first actuator within thesubset for all admissible future states of the reference trajectory; andcontrolling the first actuator according to the time-based referencetrajectory.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1A is an isometric view of a redundant laser processing machineaccording to one embodiment of an invention;

FIG. 1B is a representation of the operating range of the laserprocessing machine according to one embodiment of an invention;

FIG. 2A is a block diagram of a controller and laser processing machineaccording to embodiments of the invention;

FIG. 2B is a block diagram of the controller according to embodiments ofthe invention;

FIG. 3 is a timing diagram of machine motions according to embodimentsof the invention;

FIG. 4 is a diagram of a model of the reference trajectory andconstraints on a transient of the reference trajectory according toembodiments of the invention;

FIG. 5 is an example of a two-dimensional projection of the feasibleregion form by various constraints according to embodiments of theinvention;

FIG. 6 is a block diagram of a method for controlling an operation of amachine according to a model of a reference trajectory in accordancewith one embodiment of the invention;

FIG. 7 is a block diagram of a method for determining a controlinvariant subset according to one embodiment of the invention;

FIG. 8 is a block diagram of a test according to one embodiment of theinvention for verifying whether a polyhedral candidate subset is thecontrol invariant;

FIG. 9 is a schematic of some principles behind the determination of thecontrol invariant subset according to one embodiment of the invention;

FIG. 10 is a diagram of a method for selecting the control invariantsubset by performing iteratively a backward-reachable region computationaccording to one embodiment of the invention;

FIG. 11 is a block diagram of an alternative method that determines thecontrol invariant set;

FIG. 12 is a schematic of an effect of the method of FIG. 11 withrespect to the method of FIG. 10;

FIGS. 13 and 14 are block diagrams of method for determining a rate ofchange of the reference trajectory of a slow actuator of the laserprocessing machine according to different embodiments of the invention;

FIG. 15 is a block diagram of a method optimizing a combination of avalue of the rate of change and a volume of the subset of the feasibleregion according to some embodiments;

FIGS. 16 and 17 are schematics of some embodiments of the invention fordetermining the reference trajectory; and

FIG. 18 is a block diagram of a method according to another embodimentthat uses some principled discussed in reference with FIGS. 16 and 17.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Some embodiments of the invention provide a system and a method forcontrolling an operation of a redundant laser processing machine. Someembodiments control the machine using an optimization-based recedinghorizon control subject to constraints that guarantees the feasibilityof tracking the reference trajectory with an error defined by bounds ofa tracking error. A non-limited example of the receding horizon controlis a Model Predictive Control (MPC).

FIG. 1A shows an isometric view of an example laser processing machineaccording to one embodiment of an invention. The laser processingmachine is shown for illustration purpose and the design of this machineis not intended to limit the scope of the invention. The laserprocessing machine includes a slow actuator and a fast actuator,examples of which are provided below.

A workpiece 100 is supported on a beam dump 110 beneath a gantry 120.The gantry moves on rails 125 and 126 along a first direction, e.g.,along a Y-axis. The gantry 120 is moved along the first direction by afirst servo motor and a first screw 123. A platform 130 is arranged onthe gentry 120 and moves with the gentry along the first direction.Also, the platform 130 is moved along a second direction, e.g., along anX-axis, by a second servo motor and a second screw 135. In thisembodiment, the gentry 120, the first servo motor and the first screw123, and the second servo motor and the first screw 135 form a motionsystem for moving the platform in a plane parallel to the workpiecealong the first and the second direction. However, other embodiments ofthe invention use different types of the prismatic joints to move theplatform. For example, the first prismatic joint can include a firstdirection linear drive motor, and the second prismatic joint can includea second direction linear drive motor.

A galvano-assembly, e.g., a two-axis galvano scan head having twoorthogonal galvano drives, i.e., a first drive 140 and a second drive145, a first mirror 141 and a second mirror 146, is arranged on theplatform 130. A third motion of the first mirror 141 caused by the firstdriver 140 positions the laser beam along a third direction, and afourth motion of the second mirror 146 caused by the second driver 145positions the laser beam along a fourth direction.

In the context of this description, the gantry 120 is a first actuator,or the slow actuator, with large operating range, and the galvanoassembly is a second actuator, or the fast actuator, with smalleroperating range. However, such usage is not intended to limit the scopeof the claims. For example, in some variations the first actuators isthe fast actuator, and the second actuator is the slow actuator.

In various embodiments, the galvano assembly is arranged on the platformsuch that the third direction is fixed with respect to the firstdirection, and the fourth direction is fixed with respect to the seconddirection. For example, in one embodiment, the first direction coincideswith the third direction, and the second direction coincides with thefourth direction. In another embodiment, the first direction forms anangle of 45 degrees with the third direction, and the second directionforms the angle of 45 degrees with the fourth direction.

The galvano assembly can be affixed to the platform in order to fix thedirection of motion. Alternatively, the galvano assembly can be arrangedon the platform rotationally, such that the mutual orientations of thefirst, the second, the third, and the fourth directions can be fixedbefore, or during the operation of the laser processing machine. In thecontext of this invention, the galvano assembly is the second stage, orfast stage, with small operating range.

The laser processing machine can include a laser 150 for directing acutting laser beam 160 to the first 141 and the second 146 mirrors ofthe galvano assembly via an optical fiber 170 and a collimator 175. Inan alternative embodiment, the laser beam is directed to the galvanoassembly via diagonal mirrors moving along the Y-gantry and X-axisplatform. However, other variations are also possible.

The collimated cutting laser beam 160 is directed by the mirrors througha focusing module 180 for focusing the laser beam on the workpiece,producing a combined X-axis and Y-axis galvano assembly scan area 165 onthe workpiece 100, and cutting the workpiece 100. An example of thefocusing module 180 is a field-flattening F-theta lens or anon-telecentric F-theta lens. A size of the workpiece 100 can be greaterthan the galvano scan area 165 due to the motion of the platform.

In some embodiments, the control module includes a computer numericalcontrol (CNC) controller 195. Other embodiments can use different typesof controllers. The control module may control the motion system and thegalvano assembly according to precomputed G-code 190 that defines atrajectory of positions of the laser beam or can performs thecomputations to decide how to control the machine. For example, thecomputations can define successive positions for the X-axis platform130, the Y-axis gantry X-motion galvano assembly and mirror 141, andY-motion galvano assembly and mirror 146.

In general the machines are built with actuators that have differentdynamical behaviors. For example, the first actuator is in usuallysignificantly slower than the second actuator, due to the difference inthe displaced mass. From this difference, the indicated names of slowand fast actuators are derived.

FIG. 1B shows the relation between the position of the laser beam andthe positions achieved by the slow and fast actuation. Given the globalcoordinate frame 199, the global position 107 of the laser beam 101 isdetermined based on the position of the slow actuator 102, and therelative position of the fast stage 115 in the relative coordinate frame104 centered at the position of the slow actuator 102. The area ofreachable position for the fast actuator and for the laser beam isbounded by the area 103. Following a movement 106 for the position 112of the slow actuator, the area of reachable position for the fastactuator and for the laser beam is 113. Following a concurrent movement119 of the fast actuator, the laser beam is not at the position 118, butis in the position 111, with relative position 115 in frame 114 centeredon the position of the slow actuator and a position 117 in the globalcoordinate frame 199.

Thus, the machine operating range at any time is centered at the currentposition of the slow actuator, and has size equal to the size of theoperating range of the fast actuator. By changing the position of theslow actuator, the current machine operating range is also changed, thusrealizing an overall operating range, which is the composition of theoperating range of the slow actuator and the operating range of the fastactuator.

Some embodiments of the invention consider constraints of the slow andfast actuators of the laser processing machine in determining trajectoryand controlling the operation of the machine. For the purpose of theclarity of this disclosure, the laser processing machine with redundantactuator is arranged such that the slow actuator has larger operatingrange, but smaller velocity and acceleration limits than the fastactuator, while the fast actuator has smaller operating range, butlarger velocity and acceleration limits than the slow actuator.

Tracking Using Control Invariant Sets

Some embodiments of the invention provide a system and a method forcontrolling an operation of the laser processing machine according to amodel of a reference trajectory. Some embodiments control the machineusing an optimization-based receding horizon control subject toconstraints that guarantees the feasibility of tracking the referencetrajectory with a bounded error. A non-limited example of the recedinghorizon control is a model predictive control (MPC).

FIG. 2A shows an example machine 207, such as laser processing machine,connected to controller 205, e.g., the MPC controller, according to someembodiments of the invention. The controller 205 is programmed accordingto a model 202 of the machine. The model can be a set of equationsrepresenting changes of a state 221 and output 203 of the machine 207over time as functions of current and previous inputs 211 and previousoutputs 203. The model can include constraints 204 that representphysical and operational limitations of the machine.

During the operation, the controller receives a command 201 indicatingthe reference behavior of the machine. The command can be, for example,a motion command. In response to receiving the command 201, thecontroller generates input u 211 for the machine. In response to theinput, the machine updates the output y 203 and the state x 221 of themachine.

FIG. 2B shows a block diagram of the controller 205 according oneembodiment of the invention. The controller 205 includes a processor 291connected to a memory 292 for storing the model 202 and the constraints204, such as constraints of the machine, e.g., physical andspecification constraints, constraints on a transient of the referencetrajectory and constraints on bounds of a tracking error. For example,the constraints on the transient of the reference trajectory can includea possible type of changes of the state of the reference trajectory,rates of the change of the state of the reference trajectory amongothers. The bounds of the tracking error can include the alloweddifference between a function the state of the machine and a function ofthe state of the reference trajectory. The function can be, e.g.,identity function, linear combination.

FIG. 3 shows a timing diagram of the operation, e.g., motions, of themachine 207 according to some embodiments of the invention. Thecontroller 205 generates the input 211 for the machine to perform thereference operation while enforcing the constraints using the expectedmachine behavior according to the model. The controller at each time k301, solves a finite time optimal control problem for a predictioninterval 304, e.g., from the current time till the N next times. Theconstraints can include constraints of the machine including regions offeasible states and outputs 302 and feasible input 303.

Some embodiments of the invention are based on a realization that thelaser control problem can be expressed as a bounded tracking problem ofonly one actuator, e.g., the slow actuator. For example, the slowactuator needs to track the processing pattern with an error smallerthan an operation range of the fast actuator while enforcing theconstraints of the slow actuator. Some embodiments solve this trackingproblem by controlling the operation of the slow actuator within asubset of a feasible region of the states of the slow actuator of thelaser processing machine and states of the reference trajectory, whereinthe subset of the feasible region is control invariant with respect toall possible future values of the reference trajectory.

One embodiment model the dynamics of the slow actuator as

$\begin{matrix}{{{p\left( {k + 1} \right)} = {{p(k)} + {T_{s}{v(k)}}}}{{{v\left( {k + 1} \right)} = {{v(k)} + {\frac{T_{s}}{J}\left( {{L\; \tau} - {\beta \; {v(k)}}} \right)}}},}} & (1)\end{matrix}$

where p is a position of the slow actuator, v is a velocity of the slowactuator, τ is a torque of the slow actuator, T_(s) is a control periodof the machine at which a control cycle is executed, k is an index ofthe control cycle, J is an inertia of the slow actuator, L is a lengthof a pitch of a ball screw which converts the longitudinal motion intolinear notion, and τ is the torque of the slow actuator, β is a frictioncoefficient determining the friction torque on the slow actuator for agiven angular velocity of the slow actuator.

In general, parameters p, v, t are two dimensional vectors with x and ycoordinates, and are subject to constraints

p _(min) ≦p(k)≦p _(max)

v _(min) ≦v(k)≦v _(max)

a _(min) ≦a(k)≦a _(max)

τ_(min)≦τ(k)≦τ_(max)  (2)

that define lower and upper bounds on position p, velocity v,acceleration a, and torque τ.

One embodiment expresses the model 202 of the slow actuator as a lineardifference equation

x(k+1)=Ax(k)+Bu(k)

y(k)=Cx(k)  (3)

where k is a time instant when the signals are sampled, i.e., the indexof the control cycle, u is the machine input, y is the machine output, xis the state of the machine, and A, B, C, are parameters of the model.For example, x=[p, v]′, y=p, u=τ, and A, B, C are matrices ofappropriate dimensions, and the operation of the slow actuator issubject to linear constraints

x(k)εX,u(k)εU,∀k≧0  (4)

where X, U are polyhedral sets.

Some embodiments determine the model of the reference trajectory havinga position and a rate of change of the reference trajectory as states ofthe model or inputs to the model. Such models can represent all possiblefuture reference trajectories and the constraints on those trajectoriesneeded for the determination of the control invariant subset. Forexample, one embodiment models the reference trajectory r as aconstrained integrator of the rate of change of the reference trajectoryγ according to

r(k+1)=r(k)+y(k)

d(k)=r(k)  (5)

subject to constraints

r _(min) ≦r(k)≦r _(max).  (6)

The rate of change can be unknown but bounded

γ_(min)≦γ(k)≦γ_(max),  (7)

where the bounds on γ (γ_(min), γ_(max)) can be selected, e.g., asdescribed below.

Thus the reference trajectory can be modeled as a linear system

r(k+1)=A _(r) r(k)+B _(r)γ(k),

d(k)=C _(r) r(k),  (8)

where rεR is a reference state of the reference model and d is areference output of the reference model, γεΓ is the reference input, andA_(r), B_(r), C_(r), are the parameters that define the model of thereference trajectory. For instance for the constrained integrator modelthe reference trajectory on each axis s defined by A_(r)=1, B_(r)=1,C_(r)=1.

It is realized that the dynamics of the constrained integrator model in(5) is the simplest model that allows representing time trajectoriesthat generate any spatial laser processing pattern. Hence (5) is themodel that requires minimal complexity in terms of parameter selectionand computations and still allow processing any processing pattern.However, in other embodiments of this invention different models for thereference trajectory are developed according to (8), by selecting thevalues of A_(r), B_(r), C_(r).

The enforcement of the constraints on the fast actuator results in theconstraint connecting the linear system in (3) and the reference systemin (5) as

F _(min) ≦p(k)−d(k)≦F _(max)  (9)

where F_(max), F_(min) are the maximal and minimal value for theposition of the fast actuator generated achieved during operation of themachine. Some embodiments allow values of F_(max), F_(min) to be afraction of the actual physical range of the fast actuator, for instancehalf of the range, or a third of the range.

The bounds on the rate of change γ of the reference trajectory determinehow fast the reference position for the laser beam is allowed to move.If the rate of change γ is small, then the laser processing is slow. Ifthe rate of change γ is large, then the laser processing is faster.

For dynamics in equations (3) and (5), subject to constraints inequations (2), (6), (9) and with unknown bounded rate of change γsatisfying the equations (7), a set of constraints

H _(x) x(k)+H _(r) r(k)≦H _(c),  (10)

can describe the machine states and reference states with dynamicsaccording to equations (3) and (5) such that every possible future valueof the reference trajectory that satisfies equations (6) and (7) canalways be tracked within the range (9) of the fast actuator while alwayssatisfying constraints (4).

From equation (10), the constraints on the control command for givenstate of the machine state and the reference trajectory are

M _(x) x(k)+M _(r) r(k)+M _(u) u(k)≦M _(c),  (11)

which describes the commands such that every future reference thatsatisfies equations (5), (6), (7) can be tracked within the range of thefast actuator in equations (9), while satisfying constraints in equation(4) from the current state and reference state that satisfy theconstraints in equation (10), while also guaranteeing that constraintsin the equations (10) will be satisfied in the future.

FIG. 4 shows a diagram of the uncertainty of the model of the referencetrajectory and the constraints on the rate of change of the referencetrajectory. For example, for some iterations, such as a currentiteration at a current time 410 for a prediction horizon 415, thereference trajectory 420 can be known. But for some other iterations,e.g., for a next iteration at a time 430 for a next prediction horizon435, the reference trajectory at a step 425 may not be known.

The model of the reference trajectory constrains a range 440 of possiblefuture reference motions, thus providing the constraints on transientsof the reference trajectory. The constraints can be determined inadvance, but also can vary in dependence to previously determinedtrajectory. For example, the constraints on the range of the referencetrajectory at a specific time can be determined by a region 450.However, when the trajectory is determined at a step 425, theconstraints on the range of the reference trajectory can be tightened toa region 455. Thus, at the specific iteration, the region 455 definesthe constraints on the range of the reference trajectory.

The constraints on the state of the machine, constraints on thereference trajectory and the constraints on a range of motion of a fastactuator of the laser processing machine form a feasible region of astate of the machine and a state of the reference trajectory.Accordingly, it is possible to optimize the control maintaining thestate of the machine and the state of the reference trajectory withinthat feasible region and, thus, to convert the tracking problem into apurely optimization problem.

FIG. 5 shows an example of a two-dimensional projection of the feasibleregion 510 defined by various constraints on the operation of the laserprocessing machine according to embodiments of the invention. Due to thenature of receding horizon control, the existence of a solution for acertain horizon does not by itself guarantees the existence of thesolution for a subsequent horizon. For example, the state of the machineand a state of the reference trajectory 520 can be optimal and feasiblefor one iteration, but all control actions 521-524 that controller isallowed to take during the next iteration can bring a state of themachine outside of the feasible region 510.

Some embodiments of the invention are based on yet another realizationthat it is possible to select a subset 515 of the feasible region, suchthat from any state of the machine within that subset, there is acontrol action maintaining the state of the machine within the subsetfor the known future states of the reference trajectory or for alladmissible future states of the reference trajectory. For example, forany state such as a state 530 within the subset 515 and within allpossible control actions 531-534 that the controller can execute, thereis at least one control action 534 that maintains the state of themachine and reference within the subset 515.

Accordingly, if a control action for controlling the operation isselected such that the state of the machine remains in that specialsubset 515 of the feasible region, and the feasible region is generatedalso according to Equation (4), (6), (7) then there is a guarantee thatthe resulting optimal trajectory tracks the reference trajectory withthe bounded error and every future state of the machine always admits atleast one feasible control action. In this case, the subset 515 is acontrol invariant set. For example, the selection of the control actioncan be performed by optimizing a cost function representing theoperation of the machine subject to constraints defined by that specialsubset 515 of the feasible region, as contrasted with the optimizationwithin the feasible region 510.

FIG. 6 shows a block diagram of a method for controlling an operation ofa machine according to a model of a reference trajectory in accordancewith some embodiments of the invention. The method can be implemented ina processor 291 of the controller 205.

The method determines 610 a feasible region 510 of a state of themachine and a state of the reference trajectory defined by constraints204 including constraints of the machine, constraints on transient ofthe reference trajectory and constraints on a range of motion of a fastactuator of the laser processing machine. Next, the method selects 620 asubset 515 of the feasible region, such that from any state of themachine within the subset, there is a control maintaining the state ofthe machine within the subset and selects 630 a control action 640 forcontrolling the operation such that the state of the machine remains inthe subset. In one embodiment, the selecting includes optimizing a costfunction 635 representing the operation of the machine subject toconstraints defined by the subset region 515. The cost function isoptimized iteratively for a fixed time horizon to produce a controlaction 640 of a current iteration.

In some embodiments, all steps of the method of FIG. 6 are executedduring the operation of the machine. In alternative embodiments, thesteps 610, 620 are executed before the machine operation, and the costfunction 635 or other principles of the selection of the control actionare also defined before machine operation. Such operations can beexecuted either in a microprocessor or in a general purpose computingmachine, such as a desktop computer, laptop, or engineering workstation.The results of step 620 and the cost function 635 are programmed in thememory 292 of the microprocessor 291, and the step 630 is the only stepthat is repeatedly executed during operation of the machine.

Determining Control Invariant Subsets of Feasible Region

Some embodiments determine the set 515 as a control invariant set forthe machine model (3) and reference model (5) subject to constraints(4), (6), (7), (9). In some variations, when the reference input γ isunknown, the set 515 is a control invariant set for the machine andreference of Equations (3), (5) subject to constraints given byEquations (4), (6), (7), (9) for disturbances γ in the set Γ of thereference inputs.

FIG. 7 shows a block diagram of a method for determining the set 515according to one embodiment. The method generates 710 a number ofcandidate sets as subsets of the feasible region 510. A candidate subset720 is selected from the feasible region, and tested 730 to verifywhether the candidate subset is a control invariant. If yes 733, thecandidate subset is selected as the subset 515 and the methodsterminates. Otherwise, if another candidate subset is available 740, thenext candidate subset is selected and the test is repeated. Otherwise750, new candidates are generated 710.

FIG. 8 shows a block diagram of a test according to one embodiment ofthe invention for verifying whether a polyhedral candidate subset 720 iscontrol invariant. The method computes 810 the vertices of thepolyhedron candidate set 720 and computes 820 the vertices of polyhedralset Γ 825 of the reference inputs. Then, a vertex of the candidate setis selected 830, and checked 840 whether there exists a control action usuch that for every vertex of Γ the constraints in Equations (4), (9),are satisfied. This step can be implemented by search or mathematicalprogramming. If such control action u cannot be found, then the set isnot control invariant 845. If such control action can be found and thereare no more vertices to be checked 850, then the set is robust controlinvariant 855. Otherwise, the next vertex is selected.

However, the generation and verification of candidate sets can take asignificant amount of time, and one embodiment uses an automatedprocedure to generate and verify the candidate sets.

Because the subset 515 is a control invariant set, a method for controlinvariant set computation can be used for generating the set 515. Such amethod can be modified with the capability of enforcing constraints inEqn. (5) and considering the input γ to the model of the reference (5)as source of uncertainty.

FIG. 9 illustrates the principles behind the determination of the subset515 according to one embodiment of the invention. The feasible set 510of the state of the machine state and the state of the referencetrajectory is

X _(x,r) ={[x′r′]:xεX,∥y−d∥≦ε,rεR}.  (12)

The state of the machine 902 and the state of the reference trajectory903 define a point in (x,r) 901 in the feasible region 510. Given theadmissible set 904 for future states of the reference trajectoryaccording to

C _(γ)(r)={γ:A _(r) r+B _(r) γεR},  (13)

the future state of the machine and the future state of the referencetrajectory can be anywhere in the polyhedron 907, delimited by segments905, 906. Thus, the controller can take any control action such that thefuture state of the machine 902 remains in the segment 910, so that thecombination of the future machine state and future reference stateremains in the polyhedron 907.

If the set C _(x,r) 907 is such that there is always a control actionthat keeps the machine state and reference state 901 in the set 907 forthe entire admissible range of the reference 904

C _(x,r) ={[x′r′]′εC _(x,r) :∃uεU,Ax+BuεX,∥C(Ax+Bu)−C _(γ)(A _(γ) r+B_(γ)γ)∥≦ε,∀γεC _(γ)},  (14)

it is always possible to guarantee

[(Ax+Bu)′(A _(r) r+B _(r)γ)′]′ε C _(x,r),

which guarantees that the constraints on the machine and on the trackingerror bounds are satisfied. Notably, this procedure can be repeated at anext step because the update states are still inside C _(x,r) hencerecursively guaranteeing the constraints enforcement.

FIG. 10 shows a diagram of a method for selecting the subset C _(x,r) byperforming iteratively a backward-reachable region computation until atermination condition is satisfied. The backward-reachable regioncomputation removes all states from the current feasible region forwhich there is no control that maintains the state of the machine withinthe feasible region for reference trajectories satisfying theconstraints on the transient of the reference trajectory.

For example, the backward-reachable region computation initializes 1001at a step k=0 a current feasible region 1001 as the feasible region 510,

Ω₀ ={[x′r′]εX _(x,r)}

then determines 1002 a backward-reachable region of the states of themachine and reference trajectory that can be transition to the currentfeasible region for all the value of the admissible reference inputaccording to

Ω_(k+1) =Pre(Ω_(k) ,C _(γ)(r))={[x′r′]εX _(x,r) :∃UεU,(Ax+Bu,A _(r) r+B_(r)γ)εΩ_(k) ,∀γεC _(γ)(r)}  (15)

The computation of Equation (15) removes the states of the backwardreachable region for which there exists no control action that keepsstates into the current feasible region for all the admissible values ofthe future state of the reference.

At step 1003 the computation tests if the backward-reachable region isequal to the current feasible region, i.e., Ω_(k+1)=Ω_(k). If yes 1004,the backward-reachable region is control invariant 515 and thecomputation stops. Otherwise 1005, the backward-reachable region is usedas current feasible region in the next iteration (k=k+1) of thecomputation.

The backward-reachable region computation of FIG. 10 returns the largestexisting control invariant set. However, because the set of admissiblereference inputs is state dependent, the control invariant set is not asingle convex polyhedron but a set of convex polyhedra, which makes theuse of such subset in a real time control difficult. Also, thebackward-reachable region computation of FIG. 10 can take significantamount of time (days, weeks, months).

FIG. 11 shows a block diagram of an alternative method that determines acontrol invariant set which is slightly smaller than the largest subset,but is a single convex polyhedron. Also, the method of FIG. 11 is muchfaster, e.g., can determined the control invariant subset within minutesand/or hours.

The method initializes 1101 the matrices M, L, and the sets F, X, as

k=0,M ₀ =C,L ₀ =C _(r) ,F ₀ ={e:∥e∥≦ε},X ₀ =X

and the current feasible region

Ω₀ ={[x′r′]εX _(x,r) :rεC _(r) ^(∞)(Cx−C _(r) r)εF ₀)}

Initially, the current feasible region is the feasible region. Themethod is performed iteratively, such that the states that cannot bemaintained in the current feasible region for all the admissible valuesof the future disturbance are removed, thus forming a new currentfeasible region. The current feasible region is reformulated 1102 aspolyhedral invariant subset of feasible region, in details as anintersection of a set of the states of the machine satisfying theconstraints of the machine, the set of states on the trajectorysatisfying some constraints on the transient of the referencetrajectory, and the set of the states satisfying the error boundsconstraints, such that a forward set Ωk in the form

Ω_(k) ={[x′r′]εX _(x,r) :rεC _(r) ^(∞) ,rεX,(M _(k) x−L _(k) r)εF _(k)},

is obtained, where the constraints rεC_(r) ^(∞) the maximum outputadmissible set for the reference trajectory state that is the largestset of reference trajectory states for which the reference state admitsinput for which all the constraints (6), are satisfied. To that end,checking the constraints on the state of the machine and the constraintson the state of the reference trajectory are decoupled and the fullrange of reference input allowing the reference trajectory to move froman admissible state, regardless if the future state of the referencetrajectory is admissible, is accepted.

Then, the tightened backward-reachable region is determined 1103according to

Ω_(k+1) ={[x′r′]εX _(x,r) :∃uεU,rεC _(r) ^(∞) ,Ax+BuεX _(k),(M_(k)(Ax+Bu)−L _(k)(A _(r) r+B _(r)γ))εF _(k),∀γεΓ}.

If the tightened backward-reachable region equals 1104 the currentfeasible region, i.e.,

Ω_(k+1)==Ω_(k),

then 1105 the tightened backward-reachable region is control invariant515 and the algorithm stops with

C _(x,r)=Ω_(k).

Otherwise 1106, the matrices M, L are updated as

M _(k+1) =M _(k) A,L _(k+1) =L _(k) A _(r)

and 1107 the sets F, X are updated as

X _(k+1) ={xεX:Ax+BuεX _(k)}

F _(k+1) ={v:∃uεU,∃bεF _(k) ,v=b−M _(k) Bu+L _(k) B _(r)γ,∀γεΓ}.

Then, the tightened backward-reachable region is assigned as new currentfeasible region 1108, and a new iteration is performed from 1102 withk=k+1.

Because in the method of FIG. 1I the variable γ is now independent ofthe variable r, the resulting set C_(x,r) is a polytope.

FIG. 12 shows an effect of the method of FIG. 11 with respect to themethod of FIG. 10. Because the dependence of the reference input γ onthe reference state r is ignored, the subset 1201 obtained by the methodof FIG. 11 is smaller than the set 1202 obtained by the method of FIG.10, but the subset 1201 is a single convex polyhedron, while the subset1202 includes a larger convex polyhedron 1203 and multiple smallerconvex polyhedral 1204 at the borders, the union of which is not convex.

Optimizing Rate of Change of the Reference Trajectory

Some embodiments of the invention determine iteratively the subset ofthe feasible region and the constraints of the reference trajectoryoptimizing a processing speed of the slow actuator. The bounds of therate of change γ, i.e., (γ_(min), γ_(max)) in equation (7) determineshow fast the slow actuator can move the reference position for the laserbeam. If the bounds of the rate of change force γ to be too small, thereference position does not move much and the laser processing is slow.If the bounds of the rate of change allow γ to be large, the laserprocessing is faster, If the bounds of the rate of change allow γ to betoo large, the subset of the feasible region can be empty. Accordingly,some embodiments select a highest rate of change of the referencetrajectory resulting in a nonempty subset of the feasible region.

FIG. 13 is a block diagram of a method for determining the bounds of therate of change of the reference trajectory according to one embodimentof the invention. The method initializes 1301 the rate of change to avalue greater than zero. Usually, this value is small and γ_(min) isinitialized to −γ_(max). Then, the method determines 1302 the subset ofthe feasible region for the value of the rate of change of the referencetrajectory, using, e.g., a method of FIG. 11. The volume of the subsetis tested 1303, and the subset is a non-empty, then rate of change isstored 1304. In one variation, the new value of the rate of changereplaces any previously stored value. In another variation, all valuesare stored forming a set of pairs of values of the rate of change andvolumes of the subset of the feasible region.

Next, the method increases 1304 the value of the rate of change of thereference trajectory and repeats 1315 the determining the subset and theincreasing the value until 1316 the subset of the feasible region isempty. In such a manner, the last stored value 1306 of the rate ofchange is the highest rate of change of the reference trajectoryresulting in the nonempty subset of the feasible region.

FIG. 14 shows a block diagram of a method for determining the rate ofchange of the reference trajectory according another embodiment of theinvention. In this embodiment, the method selects 1401 a value of therate of change of the reference trajectory resulting in an empty 1403subset of the feasible region; and decreases 1404 iteratively the valueof the rate of change of the reference trajectory until the decreasedvalue results 1402 in a nonempty subset of the feasible region. In sucha manner, the last stored value 1406 of the rate of change is thehighest rate of change of the reference trajectory resulting in thenonempty subset of the feasible region.

The increase 1305 and/or the decrease 1404 can be performed in a varietyof ways, for instance by adding/removing a constant term to the currentvalue of the rate of change γ_(max) or by adding a term related to thecurrent value of γ_(max).

FIG. 15 shows a block diagram of a method optimizing a combination of avalue of the rate of change and a volume of the subset of the feasibleregion according to some embodiments. This embodiment is based onrecognition that the high rate of change of the reference trajectory canbe balance with a volume of the subset of the feasible region.

The method determines a set of pairs of values of the rate of change andvolumes of the subset of the feasible region. For example, such a setcan be determined during an execution of the methods of FIG. 13 or 14that maintain all the previously found values of γ_(max) in {γ_(max)⁽¹⁾, γ_(max) ⁽²⁾, . . . , γ_(max) ^((k))} and the corresponding controlinvariant sets {C_(x,r) ⁽¹⁾, C_(x,r) ⁽²⁾, . . . , C_(x,r) ^((k))}. Next,the method selects 1503 from the set a pair optimizing a combination ofa value of the rate of change and a volume of the subset of the feasibleregion. In one embodiment, the optimization also selects 1502 a positivescalar ω defining the relative importance of processing speed versusother objectives such as limiting accelerations and sudden movements,energy consumption, etc.

Then value of, is determined 1503 by solving an optimization problem

γ_(max) = γ_(max)^((i)):i = arg  max_(i)V(C_({x, r})^(i)) + ω γ_(max)^(i)}s.t.  i = 1, …  , N_(k)

where N_(k) is the total number of non-empty control invariant set foundin 1503, and V(C) is a measure of the size of the set C, such as thevolume, possibly scaled, the surface, or the maximum inscribed circle.The selected value of γ_(max) is used in equation (7).

Selection of Reference Trajectory

In laser processing machine applications, the reference trajectory isusually defined as points in space, i.e., a space-based trajectory, andfor the application of controls, a time-based trajectory needs to bedetermined. Hence, some embodiments are based on a realization that therate of change of the reference trajectory can be used for generatingthe time-based reference trajectory from the space-based trajectory.

Some embodiments determine the reference trajectory for the slowactuator as a function of time such that the reference trajectorysatisfies the rate of change of the reference trajectory. For example,one embodiment applies a rate bounding filter to the processing patternsuch that a change of position of a laser point within a control periodof the machine is less than the rate of change of the referencetrajectory.

The reference trajectory is a time-based reference trajectory of theslow actuator tracking the processor pattern with an error bounded by arange of motion of the fast actuator of the laser processing machine,such that a control action that changes a state of the slow actuatoraccording to the reference trajectory maintains the state of the slowactuator within a subset of the feasible region of states of the slowactuator and states of the reference trajectory.

Some embodiments of the invention determine a future feasible referencepoint for a current reference point. The feasible point is a referencepoint on the reference trajectory of the slow actuator, such that theslow actuator controlled to that reference point satisfies theconstraints in the equations (5), (6) for a value of the rate of changethat satisfies bounds in the equation (7), and that all the referencepoints between the current reference point and the future referencepoint are at a distance lesser than the size of the range of the fastactuator, and that enough time in one control sampling period isavailable for the fast actuator to process all intermediate points onthe processing pattern.

FIGS. 16 and 17 show some embodiments of the invention for determiningthe reference trajectory. The feasible reference points 1601 can befound according to the range 1603 of the fast actuator from currentreference point 1610 and according to range of achievable motions 1604as defined by equations (5), (6), and (7). The point 1620 is the firstinfeasible point outside of the range 1603 and/or the range 1604. Thepoint 1621 is the last feasible next reference point.

Accordingly, some embodiments determine 1710, for a current referencepoint on the reference trajectory, a future feasible reference point,e.g., the point 1621, satisfying the constraints on the referencetrajectory and a future infeasible point, e.g., the point 1620 violatingthe constraints on the reference trajectory. Next, the embodimentdetermines 1720 an interpolation segment 1602 connecting the futurefeasible 1621 and the future infeasible 1620 reference points, anddetermines 1730 a point 1630, furthest from the feasible point 1621 onthe interpolation segment 1602 satisfying the constraints. In otherwords, the point 1630 can be identified as the closest point to thepoint 1620 in the segment 1602 such that the point is covered by therange 1605 of the fast actuator at current reference point, is in therange of achievable motions 1604, the current reference point 1610 is inthe range of the 1605 of the fast actuator centered at the point 1630,all intermediate points 1601 are covered by ranges 1603 and 1605, andthere is enough time for the laser to operate all the intermediatepoints 1601. The point 1630 is a next reference point of the referencetrajectory.

FIG. 18 shows a block diagram of a method according to anotherembodiment that uses some principled discussed in reference with FIGS.16 and 17. The feasible point 1801 is initialized as the currentreference point, and the infeasible point 1802 is initialized as thesuccessive closest reference point. If 1803 the infeasible point is afeasible next reference point, then 1804 the feasible point value isupdated to the current value of the infeasible point and the infeasiblepoint value is updated to the value of the successive closest point tothe current infeasible point. If the infeasible point is not a feasiblenext reference point, then the interpolating segment 1805 between thefeasible point and the infeasible point is determined, and the feasiblepoint in the interpolating segment that is closest to the infeasiblepoint is determined 1806. Such point 2507 is used as the next referencepoint at the following sampling period of the control method.

The method above provides the fastest processing that is achievable withguarantees that no constraints are ever violated in the future, for allthe references that are feasible.

The above-described embodiments of the present invention can beimplemented in any of numerous ways. For example, the embodiments may beimplemented using hardware, software or a combination thereof. Whenimplemented in software, the software code can be executed on anysuitable processor or collection of processors, whether provided in asingle computer or distributed among multiple computers. Such processorsmay be implemented as integrated circuits, with one or more processorsin an integrated circuit component. Though, a processor may beimplemented using circuitry in any suitable format.

Also, the various methods or processes outlined herein may be coded assoftware that is executable on one or more processors that employ anyone of a variety of operating systems or platforms. Additionally, suchsoftware may be written using any of a number of suitable programminglanguages and/or programming or scripting tools, and also may becompiled as executable machine language code or intermediate code thatis executed on a framework or virtual machine. Typically thefunctionality of the program modules may be combined or distributed asdesired in various embodiments.

Also, the embodiments of the invention may be embodied as a method, ofwhich an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts concurrently, eventhough shown as sequential acts in illustrative embodiments.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

We claim:
 1. A method for controlling an operation of a laser processingmachine with redundant actuators including a first actuator and a secondactuator, comprising: determining a feasible region for states of thefirst actuator and states of a reference trajectory of the firstactuator defined by constraints of the laser processing machine,constraints on the reference trajectory and constraints on a range ofmotion of the second actuator; selecting a subset of the feasibleregion, such that for any state of the first actuator and any state ofthe reference trajectory within the subset, there is an admissiblecontrol maintaining the state of the first actuator within the subset ofthe to feasible region for admissible future states of the referencetrajectory determined by according to a model of the referencetrajectory and the constraints of the reference trajectory; andselecting an admissible control action for controlling the operationsuch that the state of the first actuator remains in the subset of thefeasible region, wherein the steps of the method are performed by aprocessor.
 2. The method of claim 1, wherein the constraint on thereference trajectory includes a rate of change of the referencetrajectory, further comprising: determining the model of the referencetrajectory having a position and the rate of change of the referencetrajectory as states of the model or inputs to the model.
 3. The methodof claim 1, further comprising: determining iteratively the subset ofthe feasible region and the constraints of the reference trajectoryoptimizing a processing speed of the first actuator.
 4. The method ofclaim 3, further comprising: selecting a highest rate of change of thereference trajectory resulting in a nonempty subset of the feasibleregion.
 5. The method of claim 4, further comprising: determining thesubset of the feasible region for a value of the rate of change of thereference trajectory; increasing a value of the rate of change of thereference trajectory; and repeating the determining the subset and theincreasing the value until the subset of the feasible region is empty.6. The method of claim 4, further comprising: selecting a value of therate of change of the reference trajectory resulting in an empty subsetof the feasible region; and decreasing iteratively the value of the rateof change of the reference trajectory until the value results in anonempty subset of the feasible region.
 7. The method of claim 3,further comprising: determining a set of pairs of values of the rate ofchange and a size of the subset of the corresponding feasible region;and selecting from the set of pairs a pair optimizing a combination of avalue of the rate of change and the size of the subset of the feasibleregion.
 8. The method of claim 3, further comprising: determining,according to a processing pattern, the reference trajectory for thefirst actuator as a function of time such that the reference trajectorysatisfies the rate of change of the reference trajectory.
 9. The methodof the claim 8, further comprising: applying a rate bounding filter tothe processing pattern, such that a change of position of a laser pointwithin a control period of the machine is less than the rate of changeof the reference trajectory and processing speed limitations imposed bythe fast actuator are satisfied for the change of position of the laserpoint.
 10. The method of the claim 8, further comprising: determiningfor a current reference point on the reference trajectory a futurefeasible reference point satisfying the constraints on the referencetrajectory and a future infeasible point violating the constraints onthe reference trajectory; determining an interpolation segmentconnecting the future feasible and the future infeasible referencepoints; and determining a furthest point on the interpolation segmentsatisfying the constraints as a next reference point.
 11. The method ofclaim 1, wherein the selecting further comprises: testing each state ofthe feasible region to determine the subset.
 12. The method of claim 1,wherein the selecting further comprises: partitioning the feasibleregion into a set of subsets; and selecting the subset satisfying acontrol invariance test.
 13. The method of claim 1, wherein theselecting the subset further comprises: initializing a current feasibleregion as the feasible region; performing iteratively abackward-reachable region computation until a backward reachable regionis equal to the current feasible region, wherein, for each iteration,the backward-reachable region computation removes states with no controlthat maintains the state of the machine within the current feasibleregion for the reference trajectories satisfying the constraints on thetransient of the reference trajectory.
 14. The method of claim 13,wherein the backward-reachable region computation comprises: increasinga dimensionality of the current feasible region by combining thefeasible region with a set of admissible control actions to produce ahigher dimensional region; selecting, from the higher dimensionalregion, combinations of the state of the machine, the state of thereference trajectory and the control action such that the control actionmaintains the state of the machine within the current feasible regionfor all the reference trajectories satisfying the constraints on thetransient of the reference trajectory; and selecting unique pairs of thestate of the machine and the state of the reference trajectory, suchthat at least one machine input exists such that the state of themachine, the state of the reference trajectory and the machine inputsare in the higher dimensional region.
 15. The method of claim 1, whereinthe selecting the control action comprises: optimizing a cost functionrepresenting the operation of the machine subject to constraints definedby the subset using a model of the machine, the model of the referencetrajectory, a current state of the machine, and a current state of thereference trajectory.
 16. A controller for controlling an operation of afirst actuator of a laser processing machine according to a referencetrajectory by selecting a control action for controlling the operationsuch that a state of the first actuator for admissible future states ofthe reference trajectory remains in a subset of the feasible region ofstates of the first actuator and states of the reference trajectory,wherein the feasible region is defined by constraints of the firstactuator, constraints on the reference trajectory and constraints on arange of motion of a second actuator of the laser processing machine,and wherein the subset of the feasible region is a control invariant,such that for any state of the first actuator within the subset, thereis a control maintaining the state of the first actuator within thesubset for all admissible future states of the reference trajectory. 17.The controller of claim 16, further configured for determiningiteratively the subset and a rate of change of the reference trajectoryoptimizing a processing speed of the first actuator.
 18. The controllerof claim 17, further configured for determining the reference trajectorysatisfying the rate of change of the reference trajectory.
 19. Thecontroller of claim 18, further configured for determining the referencetrajectory such that a control action from a plurality of admissiblecontrol actions that changes a current state of the first actuator intoa next state of the first actuator according to the reference trajectorymaintains the state of the first actuator within the subset of thefeasible region.
 20. A method for controlling an operation of a laserprocessing machine according to a processing pattern, wherein the laserprocessing machine includes a first actuator and a second actuator,comprising: determining a time-based reference trajectory of the firstactuator tracking the processor pattern with an error bounded by a rangeof motion of the second actuator of the laser processing machine, suchthat a control action that changes a state of the first actuatoraccording to the reference trajectory maintains the state of the firstactuator within a subset of the feasible region of states of the firstactuator and states of the reference trajectory, wherein the feasibleregion is defined by constraints of the first actuator, constraints onthe reference trajectory and constraints on the range of motion of thesecond actuator of the laser processing machine, and wherein the subsetof the feasible region is a control invariant, such that for any stateof the first actuator within the subset, there is a control maintainingthe state of the first actuator within the subset for all admissiblefuture states of the reference trajectory; and controlling the firstactuator according to the time-based reference trajectory.